To reiterate some of the above....
The more upright you sit, the more weight you put on your lower back. If you're leaning over, you are putting more weight on your arms. This requires a little more upper body and core work.
It may not be a big deal if you're doing shorter hops and take lots of breaks, but could be counter-productive when you need to spend several days in the saddle.
Perhaps a better alternative is a recumbent bike, which puts a lot less strain on your body and stays more aerodynamic. Not as cheap as a beach cruiser though
And of course, you could hunt down someone in your area who is a top-notch fitter and has experience with touring and long distance rides. It could be some other factor affecting your back, such as saddle fore-aft position.